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Towards a unified asymptotic theory for autoregression. (English) Zbl 0654.62073

Let \(y_ 1,...,y_ T\) be generated by the model \(y_ t=ay_{t-1}+u_ t\) where \(y_ 0\) is any given random variable, the innovations \(\{u_ t\}\) satisfy some rather general moment and weak dependence conditions and \(a=\exp (c/T)\) with \(-\infty <c<\infty\). If \(c\neq 0\) then \(\{y_ t\}\) is called near-integrated.
The author derives the asymptotic distributions of \[ T^{-3/2}\sum y_ t,\quad T^{-2}\sum y\quad 2_ t,\quad T^{-1}\sum y_{t-1}u_ t\quad and\quad \hat a=\sum y_ ty_{t-1}/\sum y\quad 2_{t-1}\quad as\quad T\to \infty. \] The asymptotics of some expressions are analysed also for \(c\to \pm \infty\). The general theory is expressed in terms of functionals of a simple diffusion process.
Reviewer: J.Anděl

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F05 Central limit and other weak theorems
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